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<li><a class="reference internal" href="#">Manifold Learning methods on a severed sphere</a></li>
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  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-manifold-plot-manifold-sphere-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="manifold-learning-methods-on-a-severed-sphere">
<span id="sphx-glr-auto-examples-manifold-plot-manifold-sphere-py"></span><h1>Manifold Learning methods on a severed sphere<a class="headerlink" href="#manifold-learning-methods-on-a-severed-sphere" title="Permalink to this headline">¶</a></h1>
<p>An application of the different <a class="reference internal" href="../../modules/manifold.html#manifold"><span class="std std-ref">Manifold learning</span></a> techniques
on a spherical data-set. Here one can see the use of
dimensionality reduction in order to gain some intuition
regarding the manifold learning methods. Regarding the dataset,
the poles are cut from the sphere, as well as a thin slice down its
side. This enables the manifold learning techniques to
‘spread it open’ whilst projecting it onto two dimensions.</p>
<p>For a similar example, where the methods are applied to the
S-curve dataset, see <a class="reference internal" href="plot_compare_methods.html#sphx-glr-auto-examples-manifold-plot-compare-methods-py"><span class="std std-ref">Comparison of Manifold Learning methods</span></a></p>
<p>Note that the purpose of the <a class="reference internal" href="../../modules/manifold.html#multidimensional-scaling"><span class="std std-ref">MDS</span></a> is
to find a low-dimensional representation of the data (here 2D) in
which the distances respect well the distances in the original
high-dimensional space, unlike other manifold-learning algorithms,
it does not seeks an isotropic representation of the data in
the low-dimensional space. Here the manifold problem matches fairly
that of representing a flat map of the Earth, as with
<a class="reference external" href="https://en.wikipedia.org/wiki/Map_projection">map projection</a></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Jaques Grobler &lt;jaques.grobler@inria.fr&gt;</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <span class="n">time</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
<span class="kn">from</span> <span class="nn">matplotlib.ticker</span> <span class="kn">import</span> <span class="n">NullFormatter</span>

<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">manifold</span>
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">check_random_state</span>

<span class="c1"># Next line to silence pyflakes.</span>
<span class="n">Axes3D</span>

<span class="c1"># Variables for manifold learning.</span>
<span class="n">n_neighbors</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">1000</span>

<span class="c1"># Create our sphere.</span>
<span class="n">random_state</span> <span class="o">=</span> <span class="n">check_random_state</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">random_state</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">-</span> <span class="mf">0.55</span><span class="p">)</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">random_state</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span>

<span class="c1"># Sever the poles from the sphere.</span>
<span class="n">indices</span> <span class="o">=</span> <span class="p">((</span><span class="n">t</span> <span class="o">&lt;</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">-</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">8</span><span class="p">)))</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">t</span> <span class="o">&gt;</span> <span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">8</span><span class="p">))))</span>
<span class="n">colors</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="n">indices</span><span class="p">]</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="n">indices</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">indices</span><span class="p">]),</span> \
    <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="n">indices</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">indices</span><span class="p">]),</span> \
    <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="n">indices</span><span class="p">])</span>

<span class="c1"># Plot our dataset.</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;Manifold Learning with </span><span class="si">%i</span><span class="s2"> points, </span><span class="si">%i</span><span class="s2"> neighbors&quot;</span>
             <span class="o">%</span> <span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="n">n_neighbors</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>

<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">251</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="n">indices</span><span class="p">],</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">rainbow</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span>

<span class="n">sphere_data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">])</span><span class="o">.</span><span class="n">T</span>

<span class="c1"># Perform Locally Linear Embedding Manifold learning</span>
<span class="n">methods</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;standard&#39;</span><span class="p">,</span> <span class="s1">&#39;ltsa&#39;</span><span class="p">,</span> <span class="s1">&#39;hessian&#39;</span><span class="p">,</span> <span class="s1">&#39;modified&#39;</span><span class="p">]</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;LLE&#39;</span><span class="p">,</span> <span class="s1">&#39;LTSA&#39;</span><span class="p">,</span> <span class="s1">&#39;Hessian LLE&#39;</span><span class="p">,</span> <span class="s1">&#39;Modified LLE&#39;</span><span class="p">]</span>

<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">method</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">methods</span><span class="p">):</span>
    <span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
    <span class="n">trans_data</span> <span class="o">=</span> <span class="n">manifold</span>\
        <span class="o">.</span><span class="n">LocallyLinearEmbedding</span><span class="p">(</span><span class="n">n_neighbors</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span>
                                <span class="n">method</span><span class="o">=</span><span class="n">method</span><span class="p">)</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">sphere_data</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
    <span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">%s</span><span class="s2">: </span><span class="si">%.2g</span><span class="s2"> sec&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">methods</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>

    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">252</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">trans_data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">trans_data</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">rainbow</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">%s</span><span class="s2"> (</span><span class="si">%.2g</span><span class="s2"> sec)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">labels</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>

<span class="c1"># Perform Isomap Manifold learning.</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">trans_data</span> <span class="o">=</span> <span class="n">manifold</span><span class="o">.</span><span class="n">Isomap</span><span class="p">(</span><span class="n">n_neighbors</span><span class="p">,</span> <span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>\
    <span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">sphere_data</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">%s</span><span class="s2">: </span><span class="si">%.2g</span><span class="s2"> sec&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="s1">&#39;ISO&#39;</span><span class="p">,</span> <span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>

<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">257</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">trans_data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">trans_data</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">rainbow</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">%s</span><span class="s2"> (</span><span class="si">%.2g</span><span class="s2"> sec)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="s1">&#39;Isomap&#39;</span><span class="p">,</span> <span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>

<span class="c1"># Perform Multi-dimensional scaling.</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">mds</span> <span class="o">=</span> <span class="n">manifold</span><span class="o">.</span><span class="n">MDS</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_init</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">trans_data</span> <span class="o">=</span> <span class="n">mds</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">sphere_data</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;MDS: </span><span class="si">%.2g</span><span class="s2"> sec&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>

<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">258</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">trans_data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">trans_data</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">rainbow</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;MDS (</span><span class="si">%.2g</span><span class="s2"> sec)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>

<span class="c1"># Perform Spectral Embedding.</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">se</span> <span class="o">=</span> <span class="n">manifold</span><span class="o">.</span><span class="n">SpectralEmbedding</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
                                <span class="n">n_neighbors</span><span class="o">=</span><span class="n">n_neighbors</span><span class="p">)</span>
<span class="n">trans_data</span> <span class="o">=</span> <span class="n">se</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">sphere_data</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Spectral Embedding: </span><span class="si">%.2g</span><span class="s2"> sec&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>

<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">259</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">trans_data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">trans_data</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">rainbow</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Spectral Embedding (</span><span class="si">%.2g</span><span class="s2"> sec)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>

<span class="c1"># Perform t-distributed stochastic neighbor embedding.</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">tsne</span> <span class="o">=</span> <span class="n">manifold</span><span class="o">.</span><span class="n">TSNE</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="s1">&#39;pca&#39;</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">trans_data</span> <span class="o">=</span> <span class="n">tsne</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">sphere_data</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;t-SNE: </span><span class="si">%.2g</span><span class="s2"> sec&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>

<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">trans_data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">trans_data</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">rainbow</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;t-SNE (</span><span class="si">%.2g</span><span class="s2"> sec)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t1</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">NullFormatter</span><span class="p">())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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